The APOS linear programming solver : an implementation of the homogeneous algorithm

The purpose of this work is to present the APOS linear programming (LP) solver intended for solution of large-scale sparse LP problems. The solver is based on the homogeneous interior- point algorithm which in contrast to the primal-dual algorithm detects a possible primal or dual infeasibility reliably. It employs advanced (parallelized) linear algebra, it handles dense columns in the constraint matrix efficiently, and it has a basis identification procedure. Moreover, recently the solver has been incorporated into the commercially available XPRESS-MP software. This paper discusses in details the algorithm and linear algebra employed by the APOS LP solver. In particular the homogeneous algorithm is emphasized. Furthermore, extensive com- putational results are reported. These results include comparative results for the XPRESS-MP simplex and barrier code and the freely available BPMPD code developed by Cs. M?esz?aros. Finally, computational results are presented to demonstrate the possible speed-up, when using a parallelized version of the APOS LP solver on a Silicon Graphics Challenge computer.

A parallel interior-point algorithm for linear programming on a shared memory machine

The XPRESS interior point optimizer is an “industrial strength” code for solution of large-scale sparse linear programs. The purpose of the present paper is to discuss how the XPRESS interior point optimizer has been parallelized for a Silicon Graphics multi processor computer. The ma jor computational task, performed in each iteration of the interior-point method implemented in the XPRESS interior point optimizer is the solution of a symmetric and positive definite system of linear equations. Therefore, parallelization of the Cholesky decomposition and the triangular solve procedure are discussed in detail. Finally, computational results are presented to demonstrate the parallel efficiency of the optimizer. It should be emphasized that the methods discussed can be applied to the solution of large-scale sparse linear least squares problemslinear programming, interior-point methods, parallel computing.